Comprison of two nisotropi rp mols t lmnt lvl N. Sivsithmprm Plxis B.V, Dlft, Th Nthrlns Univrsity of Strthly, Glsgow, Unit Kingom M. Krstunn Chlmrs Univrsity of Thnology, Gothnurg, Swn Univrsity of Strthly, Glsgow, Unit Kingom R.B.J. Brinkgrv Plxis B.V, Dlft, Th Nthrlns Dlft Univrsity of Thnology, Dlft, Th Nthrlns P. G. Bonnir Plxis B.V, Dlft, Th Nthrlns. ABSTRACT: This ppr prsnts omprison of two nisotropi rp mols, ACM n Crp-SCLAY1, whih iffr in thir formultion of rp strin rt. Crp is formult in ACM using th onpt of ontours of onstnt volumtri rp strin rt, whrs th nwly vlop Crp-SCLAY1 mol uss th onpt of onstnt rt of viso-plsti multiplir. Th two mols r intil in th wy th initil nisotropy n th volution of nisotropy r simult. A ky ssumption of oth mols is tht thr is no purly lsti omin. Th mols r ompr t lmnt lvl. Th numril simultions show tht th Crp-SCLAY1 mol is l to giv ttr rprsnttion of nturl ly hviour t lmnt lvl. 1 INTRODUCTION Nturl soils hv in highly nisotropi mnnr u to th position pross n susqunt loing history, n show tim-pnnt (rp) hviour. An urt sription of nisotropy n rt-pnnt hviour of soft soils is nssry for sf n onomi sign of struturs on soft soils posits. To otin rlisti solutions for gostruturs on nturl lys, it is ssntil to us onstitutiv mol tht ounts for nisotropy n tim pnny. Mny onstitutiv mols for tim-pnny n nisotropy hv n propos in th litrtur. Tim-pnnt mols tht rprsnt only inhrnt nisotropy hv n propos (.g Skiguhi & Oht (1977) n Zhou t l. (2006)) s wll s tim-pnnt mols ounting for oth inhrnt n plsti strin inu nisotropy (.g. Loni t l. (2008) n Krstunn & Yin (2010)). Th onstitutiv mols shoul rltivly simpl n sy to unrstn. Illy, it shoul possil to trmin th vlus of th mol prmtrs from stnr lortory tsts. This woul nmly nhn th onfin of prtiing gothnil nginrs for opting th mols for numril nlysis. Th ACM (Loni t l. 2008) is n xtnsion of th Soft Soil Crp mol (Vrmr t l. 1998) with rott llipss (similr to th S-CLAY1S mol (Krstunn t l. 2005)) us s ontours of volumtri rp strin rts. Th formultion for th Crp-SCLAY1 mol ws propos rntly y Sivsithmprm (2012). Anisotropy in oth mols is sri y introuing fri tnsor to rprsnt th rottion of th onstitutiv llipss in th p q pln, similr to th S-CLAY1S mol (Krstunn t l. 2005). Morovr, rottionl hrning lw sris th volution of nisotropy u to volumtri n vitori rp strin rts. Howvr, th Crp- SCLAY1 mol iffrs onsirly from th ACM in th formultion of rp strin rts. Crp is formult in Crp-SCLAY1 using th onpt of rt of th viso-plsti multiplir (Grimst t l. 2010). Unlik Grimst t l. (2010) mol tht us Jnu s tim-rsistn onpt, th prsnt mol uss th mor fmilir rp offiint, moifi rp inx µ whih n sily riv from stnr lo-
rtory tsts. This ppr shows irt omprison of oth mols n thir prition pility t lmnt lvl. Th first prt of this ppr givs short sription of th ACM n Crp-SCLAY1 mols in trixil strss sp. In furthr stions th singl lmnt simultions rsults otin using th finit lmnt o PLAXIS (Brinkgrv t l. 2012) SoilTst fility r prsnt, follow y rif onlusions. q M CSS p' q NCS p' p p' 2 ANISOTROPIC CREEP MODELS Th lsti n rp prts in th oth mols r omin with n itiv lw xprssing th totl strin rt s omintion of lsti n rp omponnt s in lssil lsto-plstiity. () ACM ɛ = ɛ + ɛ (1) whr ɛ is strin, ot ovr symol implis rt (iffrntition with rspt to tim) n suprsripts n rfr to th lsti n rp omponnts rsptivly. For th sk of simpliity, th mthmtil formultion of th oth mols is prsnt in trixil strss sp, whih n us only to mol th tsting of ross-nisotropi smpls (ut vrtilly from th soil posit) in oomtr or trixil pprtus in th lortory. q M CSS p' q NCS p' p p' 2.1 ACM Loni t l. (2008) propos th Anisotropi Crp Mol (ACM) xtning from prviously vlop isotropi rp mol (Vrmr t l., Vrmr n Nhr ) whih is s on llipss of Moifi Cm Cly (Roso & Burln 1968). An xtrt of th mthmtil formultion from Loni t l. (2008) is prsnt low. Th outr rott llips fins th norml onsolition surf (NCS) n th siz of this llips volvs with volumtri rp strins oring to th hrning lw ( p p = p ɛ ) v p0xp λ κ (2) whr λ n κ r th moifi omprssion inx n moifi swlling inx rsptivly. Th intrstion of th vrtil tngnt to th llips with p xis is th isotropi pronsolition prssurp p (s Figur 1()). Th innr llips psss through th urrnt stt of fftiv strss ll th urrnt strss surf (CSS). Th intrstion of th CSS with th horizontl xis is ll th quivlnt mn strssp q, n it is fin s p q = p + (q αp ) 2 (M 2 α 2 )p (3) () Crp-SCLAY1 Figur 1: Currnt stt surf n norml onsolition surf in trixil strss sp whrm is th strss rtio t ritil stt n slr quntity α is us to sri th orinttion of th norml onsolition surf n urrnt strss surf. Th volumtri rp strin rt is givn y powr lw s follows: ɛ v = µ τ ( ) p β q (4) p p µ is rfrr to s th moifi rp inx,τ is ll th rfrn tim n is st to 1 y if th NCS is foun y prforming stnr 24h oomtr tst, nβ is fin s: β = λ κ µ (5) Th ACM nnot prit swlling on th ry si of ritil stt lin s it os not llow th strss stt to ross th filur lin rprsnt y Mohr- Coulom ritrion i.. llowing ɛ v 0. Bus of
this, th ACM is limit to th wt si of th ritil stt lin only (s Figur 1()). In ition, th ACM nnot giv stisftory rspons for strin rt hngs in unrin tsts of normlly onsolit lys (Grimst 2009) s isuss ltr. For furthr tils of th nisotropy n rp formultion, th intrst rr is rfrr to Loni t l. (2008), Whlr t l. (2003) n Krstunn t l. (2005). strin orrspons to Eq. (4). Grimst (2009) suggst tht rp xprss irtly on th rt of plsti multiplir givs th propr rspons. Th urrnt strss surf (CSS) n norml onsolition surf (NCS) r fin similr to ACM. Howvr, Crp-SCLAY1 prits swlling on th ry si of th ritil stt lin, unlik ACM (s Figur 1()). Figur 2 omprs th norml onsolition surf of ACM n Crp-SCLAY1S in gnrl strss sp. For furthr tils of th mthmtil formultion of th mol, th rr is rfrr to Sivsithmprm (2012). 3 MODEL PARAMETERS Both mols rquir th sm prmtrs sri low. () ACM Prmtrs whih r similr to th Moifi Cm-ly prmtrs inlu soils onstnts ν (Poisson s rtio), M (strss rtio t ritil stt), λ (moifi omprssion inx) n κ (moifi swlling inx). Furthrmor, th initil vlu for stt vril p m0 (initil siz of th yil surf) is rquir. In th ontxt of finit lmnt nlyss, th initil vlu of p m0 is lult s on th OCR (vrtil ovronsolition rtio) orpop (pr-ovrurn prssur), normlly onsolit K0 NC vlu (ltrl rth prssur t rst, stimt y Jky s formul) n th initil vrtil fftiv strss. Prmtrs sriing initil nisotropy (α 0 ) n its volution, inlu soil onstnts ω (rt of rottion of th surfs) nω (rltiv rt of surf rottion). Th slr vlu α 0 n ω n thortilly riv s on M vlus ( s Whlr t l. (2003) for tils) s follows: () Crp-SCLAY1 α 0 = η2 0 +3η 0 M 2 3 (7) Figur 2: Norml Consolition Surf (NCS) in gnrl strss sp 2.2 Crp-SCLAY1 In Crp-SCLAY1, Eq. (4) is moifi to n xprssion tht givs th rt of th viso-plsti multiplir s follows: Λ = µ τ ( ) p β ( q M 2 α 2 ) p p M 2 η 2 (6) whr η = q/p n th itionl trm (M 2 α 2 )/(M 2 η 2 ) is to nsur tht unr oomtr onitions, th rsulting rp ω = 3 4M 2 4η0 2 3η 0 (8) 8 η0 2 M 2 +2η 0 whr η 0 = 3(1 K NC 0 )/(1+2K NC 0 ). Th prmtr ω n stimt s on initil nisotropy (α 0 ), moifi omprssion inx (λ ), M n ω (s Loni t l. (2008) for tils) s follows: ω = 1 λ ln10m2 2α 0 ω M 2 2α 0 ω (9) In rivtion of Eq. (9), numr of ssumptions hs n m (s Loni t l. (2008)).
Consquntly, with rtin prmtr omintions Eq. (9) might rsults with ngtiv vlu, whih mks no physil sns. As n ltrntiv, n mpiril formul suggst y Zntr t l. (2002) to stimt thω vlu n us: 10 λ ω 20 λ (10) µ (moifi rp inx) n otin y msuring th volumtri strin on th long trm n plotting it ginst th logrithmi tim. τ (th rfrn tim, whih is link to th finition of vrtil pronsolition strss) n usully tkn to qul on y (s Brinkgrv t l. (2012) for tils). 4 NUMERICAL SIMULATION This stion isusss th prformn of oth mols in singl lmnt simultion. Both mols r implmnt into th finit lmnt o PLAXIS s usr-fin soil mols. Th Crp-SCLAY1 mol hs n implmnt y th first uthor n th ACM hs n implmnt y Loni t l. (2008). Singl lmnt simultions wr on using th PLAXIS SoilTst fility to highlight th similritis n th iffrns in th mol pritions. Prmtrs us for ths simultions orrsponing to Bothknnr ly prmtrs (Symposium 1992) r summriz in Tl 1. Tl 1: Bothknnr ly prmtrs. Prmtrs vlu λ 0.1 ν 0.2 κ 0.00667 M 1.5 OCR 1.5 α 0 (initil nisotropy offiint) 0.59 ω (nisotropy offiint).0 ω (nisotropy offiint) 1.0 µ (visosity offiint) 5.07x10 3 τ (visosity offiint) 1.0 y Firstly, Crp-SCLAY1 n ACM wr ompr in unrin omprssion simultions with two strin rts (10% pr y n 100% pr y). Initil fftiv strss σ 3 = 100 kp n K 0 = 0.5 wr ssum n 10% mximum strin ws ppli. Figurs 3() n 3() show th strss pths n vitori strss vrsus xil strin prit y th two mols. Though oth mols r l to prit pnn on strin rt, th pk unrin strngth prit y ACM is lowr thn tht prit y th Crp- SCLAY1 mol. In ontrst to Crp-SCLAY1, ACM prits strss pth prohs th with ruing p n q, onvrging towrs th strss origin u to th ssumption of onstnt volumtri rp strins. In th ACM simultions, jums wr osrv s highlight in Figur 3 u to th trnsition twn urrnt stt surf to Mohr-Coulom filur surf. Furthrmor, th ACM nnot rh to ritil stt onition with shring t onstnt volum n fftiv strsss. Svrl pulitions (.g. Grhm t l. (1983), Ttsuok t l. (2002), Tvns t l. (1978) n Vi & Cmpnll (1977) ) show th influn of stp hngs in strin rt on th strss-strin hviour of soft soil in unrin trixil omprssion. Immitly ftr n inrs in strin rt th strss-strin pth is sn to jump upwrs n show n initil stiff rspons. If th strin rt is ru k to th originl strin rt thn ownwrs strss jump is osrv ftr whih th pth rjoins th originl urv fin y th lowr strin rt. Th pths in strssstrin urvs r init to uniquly fin y th strin rt n th ffts of strin rt hngs r osrv to prsistnt, whih is hrtristi of isoth hviour, i.., thr is uniqu strss-strin strin-rt rltion for givn soil. Most soft lys in oth unistur n ronstitut stts, unistur nturl stiff lys n ss of soft rok ll show isoth visous hviour. Figur 4 shows stpwis hng in strin rt unrin omprssion simultions using Crp-SCLAY1 n ACM to vrify th pility of oth mols to prit th isoth hviour. Figur 4() lrly monstrts tht ACM nnot proprly simult th isoth hviour osrv in nturl soft lys unr stpwis hng in strin rt. Furthrmor, th strss pth simult y ACM nnot ovrpss th ritil stt s shown in Figur 4(). This too is not in grmnt with xprimntl osrvtions for slightly strutur or ronstitut lys (Yin t l. 2010). Thr is mthmtil iffrn twn th two mols to lult th rp strin omponnts in gnrl strss sp. In ACM n Crp-SCLAY1, rp strins r lult s follows: ACM: ɛ ij = ɛ v p q (11) p q σ ij p Crp-SCLAY1: ɛ ij = ɛ v ( p q p )NC p q σ ij (12) Th vlu of p q/ p in th ACM rhs to infinity whn η/m oms to 1, i.., th strss onition rhs to ritil stt (s Figur 5). This uss numril prolms. 5 CONCLUSIONS This ppr stuis th prformn of two nisotropi rp onstitutiv mols t lmnt lvl. In th
180 Crp-SCLAY1 ACM 100% y -1 180 100% y -1 1 1 10% y -1 1 120 1 155 10% y -1 1 120 100% y -1 10% y -1 135 100 1 130 125 145 85 90 95 100 105 80 90 100 110 120 130 p' [kp] () strss pth 100 0.00 0.02 0.04 0.06 0.08 0.10 Axil strin [-] () strss-strin pth Figur 3: Simultion of unrin trixil omprssion with vrying strin rt 2.0 % y -1 0.2 % y -1 0.02 % y -1 0.002 % y -1 20 30 p' [kp] () strss pth of Crp-SCLAY1 0.00 0.02 0.04 0.06 0.08 0.10 Axil strin [-] () strss-strin pth of Crp-SCLAY1 q [Kp] 2.0 % y -1 0.2 % y -1 0.02 % y -1 0.002 % y -1 20 30 p' [kp] () strss pth of ACM 0.00 0.02 0.04 0.06 0.08 0.10 Axil strin [-] () strss-strin pth of ACM Figur 4: Simultion of unrin trixil omprssion with vrying strin rt
p'q /p' 15 10 Crp-SCLAY1 ACM CREEP (Crp of Gomtrils, PIAP-GA-2011-286397) support y th Europn Community through th progrmm Mri Curi Inustry- Ami Prtnrships n Pthwys (IAPP). 5 0 0.0 0.2 0.4 0.6 0.8 /M Figur 5: p q/p vrsus η/m plot first mol ACM (Loni t l. 2008), th rp strin rt is formult using ontours of volumtri rp strin rts whrs in th nwly vlop mol, Crp-SCLAY1 (Sivsithmprm 2012), rp strin rt is formult using th onpt of rt of viso-plsti multiplir. Th mol simultions monstrt tht th nw formultion (Crp-SCLAY1) rsults in ttr prition of nturl soft soil hviour. Th following osrvtions r m from th omprison: Though oth mols r l to prit rt fft pnn in unrin omprssion simultion, in ontrst to th Crp-SCLAY1 mol, th ACM prits strss pths whih pproh th with ruing p n q, onvrging towrs th strss origin. ACM nnot rh ritil stt onition with shring t onstnt volum n fftiv strsss. Unrin omprssion using stpwis hng in strin rt simultions monstrt tht th ACM nnot rprou th isoth hviour osrv in nturl soft soils. Furthrmor, ACM nnot ovrpss th ; this my not in grmnt with xprimntl osrvtions for slightly strutur or ronstitut lys. Thr is mthmtil iffiulty in ACM. Whn lulting rp strin rts, th vlu of p q/ p n rh infinity whn η/m oms to 1, i.., th strss onition rhs ritil stt. Furthr work will involv ompring th prformn of th mols ginst xprimntl t n instrumnt tst struturs. 6 ACKNOWLEDGMENTS Th rsrh ws rri out s prt of GEO- INSTALL (Molling Instlltion Effts in Gothnil Enginring, PIAP-GA-2009-230638) n REFERENCES Brinkgrv, R., E. Engin, & W. M. Swolfs (2012). PLAXIS Finit Elmnt Co for Soil n Rok Anlyss. Th Nthrlns: 2D-Vrsion 2011. Grhm, J., J. Crooks, & A. Bll (1983). Tim ffts on th strss-strin hviour of nturl soft lys. Gothniqu 33(3), 327 3. Grimst, G. (2009). Dvlopmnt of fftiv strss s nisotropi mols for soft lys. Ph. D. thsis, Norwgin Univrsity of Sin n Thnology (NTNU), Norwy. Grimst, G., S. At, S. Norl, & M. Krstunn (2010). Moling rp n rt ffts in strutur nisotropi soft lys. At Gothni 5, 69 81. Krstunn, M., H. Krnn, S. Whlr, M. Koskinn, & R. Zntr (2005). Th fft of nisotropy n struturtion on th hviour of murro tst mnkmnt. Intrntionl Journl of Gomhnis (ASCE) 5(2), 87 97. Krstunn, M. & Z. Y. Yin (2010). Molling tim-pnnt hviour of murro tst mnkmnt. Gothniqu 29, 1 34. Loni, M., M. Krstunn, & P. Vrmr (2008). Anisotropi rp mol for soft soils. Gothniqu 58 (3), 215 226. Roso, K. & J. Burln (1968). On th gnrlis strss-strin hviour of wt ly. Enginring Plstiirv, 535 9. Skiguhi, H. & H. Oht (1977). Inu nisotropy n tim pnny in lys. 9th ICSMFE, Tokyo, Constitutiv qutions of Soils 17, 229 238. Sivsithmprm, N. (2012). Molling rp hviour of soft soils. Intrnl rport Plxis B. V. Symposium (1992). Bothknnr soft ly tst sit: Chrtriztion n lssons lrn (géothniqu symposium in print). Géothniqu 42(2), 161 380. Ttsuok, F., M. Ishihr, H. Di Bntto, & R. Kuwno (2002). Tim-pnnt shr formtion hrtristis of gomtrils n thir simultion. Soils & Fountions 42(2), 103 138. Tvns, F., S. Lrouil, P. L Rohll, & M. Roy (1978). Crp hviour of n unistur lightly ovronsolit ly. Cnin Gothnil Journl 15(3), 2 423. Vi, Y. & R. Cmpnll (1977). Tim-pnnt hviour of unistur ly. ASCE J Goth Eng Div 103(7), 693 9. Vrmr, P. A. & H. Nhr. A soft soil mol tht ounts for rp. Byon 2000 in Computtionl Gothnis, R.B.J. Brinkgrv(s), Blkm, Rottrm.. Vrmr, P. A., D. F. E. Stoll, & P. G. Bonnir. From lssil thory of sonry omprssion to morn rp nlysis. Pro. 9th Int. Conf. Comp. Mth. n Av. Gomh., Yun(s). Whlr, S., A. Näätänn, M. Krstunn, & M. Lojnr (2003). An nisotropi lsto-plsti mol for soft lys. Cnin Gothnil Journl, 3 418. Yin, Z. Y., C. S. Chng, M. Krstunn, & P. Y. Hihr (2010). An nisotropi lsti-visoplsti mol for soft lys. Int. J. of Solis n Struturs 47, 665 677. Zntr, R., M. Krstunn, C. Wiltffsky, H. F. Shwigr, & M. Koskinn (2002). Comprison of two pprohs for molling nisotropy of soft lys. Pro. 8th Int. Symp. on Numril Mols in Gomhnis (NUMOG VIII), 115 121. In G. N. Pn & S. Pitruszzk (s.), Rom. Zhou, C., J.-H. Yin, J.-G. Zhu, & C.-M. Chng (2006). Elsti nisotropi visoplsti moling of th strin-rt pnnt strssstrin hviour of k0-onsolit nturl mrin lys in trixil shr tst. Int. J. Gomh 5(3), 218 232.